PERIOD OF VIBRATION OF STEAM VESSELS. 119 



kinetic energy; 5 is the movement of the weight in feet, in the expression for the 

 work done. 



From formula ( i ) the expression for the number of vibrations per minute is 

 obtained: — 



The purpose of this investigation is to determine the above-mentioned for- 

 mulae, and how the values of k^ and of h, which should be used in them, can be 

 obtained when the period of vibration of a vessel is in question. In order to pre- 

 pare the way for this latter problem, the simpler cases are discussed at some 

 length in the Appendix. 



A vessel is a beam of varying moment of inertia of section "I," irregularly 

 loaded, and supported by fluid pressure. In order to simplify the calculations, the 

 moment of inertia "I" may be considered constant, although in some types of ves- 

 sels a considerable error would be introduced by this assumption. The curve of 

 weights is plotted from the data available, and the curve of buoyancy in still water 

 is obtained from the lines. The difference between the ordinates of these two 

 curves gives the curve of loads. The curve of "loads" is integrated graphically, 

 and the resulting curve is the "shear" curve. By integrating the "shear" curve, 

 we obtain the curve of "bending moments in still water." The process to this point 

 is that commonly in use in ship calculations. Plate 56 shows the results for the oil 

 tanker previously mentioned. 



By integration of the moment curve, we obtain curve of "angle of deflection" 

 and by integration of this curve we obtain curve of actual "deflection." The in- 

 tegration of the moment curve should preferably start at amidships and proceed 

 both ways. Plate 57 shows the resulting deflection curve rectified to show same 

 amount of deflection at stem and propeller post. If account had been taken of 

 varying moment of inertia of section, this curve would be slightly modified, but 

 probably inappreciably, as "I" hardly changes for three-quarter length amidships in 

 the vessel selected. 



The question next to be settled is the position to be assumed for the points 

 of support of the vessel considered as a beam. After giving the matter some 

 thought, it was concluded that this might be determined by obtaining the mean im- 

 mersion by taking the load-water plane of the vessel and determining the volume 

 of the buoyancy included between the "deflection" curve and the horizontal at the 

 lowest point of this curve. Dividing this result by the area of the load-water 

 plane, we obtain the mean immersion which will give the same volume. We plot 

 a line across the rectified "deflection" curve at this height above the lowest point 

 amidships. The crossings of this line and the deflection curve will represent the 

 points of support, as, if the vessel were to straighten out, this is the height at which 

 it would float, the buoyancy lost by the rising of the midship part being balanced 



